MinT - Best Known (224−135, 224, s)-Nets in Base 4

Best Known (224−135, 224, s)-Nets in Base 4

(224−135, 224, 104)-Net over F₄ — Constructive and digital

Digital (89, 224, 104)-net over F₄, using

t-expansion [i] based on digital (73, 224, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(224−135, 224, 129)-Net over F₄ — Digital

Digital (89, 224, 129)-net over F₄, using

t-expansion [i] based on digital (81, 224, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(224−135, 224, 813)-Net in Base 4 — Upper bound on s

There is no (89, 224, 814)-net in base 4, because

1 times m-reduction [i] would yield (89, 223, 814)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 193 646539 530053 930020 008383 954844 836820 869066 942839 700342 198078 037008 936892 252367 825191 495813 217521 986568 527922 967260 231588 860637 327160 > 4²²³ [i]